# Joyal's CatLab Natural transformations

**Basic Category Theory** ## Contents * Categories * Functors * Natural transformations * Functors of two variables * Adjoint Functors and Monads **Category theory** ##Contents * Contributors * References * Introduction * Basic category theory * Weak factorisation systems * Factorisation systems * Distributors and barrels * Model structures on Cat * Homotopy factorisation systems in Cat * Accessible categories * Locally presentable categories * Algebraic theories and varieties of algebras

# Contents

## Definition

• If $\mathbf{C}$ and $\mathbf{D}$ are two categories, then the functors $\mathbf{C}\to \mathbf{D}$ are the objects of a category $[\mathbf{C},\mathbf{D}]$ called the functor category. A morphism $\alpha:F\to G$ between two functors $F, G:\mathbf{C} \to \mathbf{D}$ is called a natural transformation. It assigns to every [object] $A\in \mathbf{C}$ a [morphism] $\alpha_A:F A \to G A$ in $\mathbf{D}$ such that the following diagram commutes
$\xymatrix{ F A\ar[d]_{F(f)}\ar[r]^{\alpha_A}& G A \ar[d]^{G(f)} \\ F B \ar[r]^{\alpha_B}&G B}$

for every morphism $f:A \to B$ in $\mathbf{C}$.

We shall often write $\alpha:F\to G: \mathbf{C}\to \mathbf{D}$ to indicate that $\alpha$ is a natural transformation between two functors $\mathbf{C}\to \mathbf{D}$.

The composite of two natural transformations $\alpha:F\to G$ and $\beta:G\to H$ in the category $[\mathbf{C},\mathbf{D}]$ is the natural transformation $\beta \alpha:F\to H$ obtained by putting $(\beta\alpha)_A=\beta_{A}\alpha_A$ for every object $A\in \mathbf{C}$,

$\xymatrix{F A\ar[d]_{F(f)}\ar[r]^{\alpha_A}& G A \ar[d]^{G(f)} \ar[r]^{\beta_A}& H A \ar[d]^{H(f)} \\ F B \ar[r]^{\alpha_B}&G B\ar[r]^{\beta_B}&H B.}$

#### Remark

The category $[\mathbf{C},\mathbf{D}]$ is locally small if $\mathbf{C}$ is small and $\mathbf{D}$ locally small, in which case we shall often denote it by $\mathbf{D}^{\mathbf{C}}$. The category $\mathbf{D}^{\mathbf{C}}$ is small, if additionally $\mathbf{D}$ is small.

## Examples

• Recall that a [presheaf] on a small category $C$ is defined to be a [contravariant functor] $C\to \mathbf{Set}$, or equivalently a covariant functor $C^o\to \mathbf{Set}$. If $X$ and $Y$ are presheaves on $C$, then a map $X\to Y$ is defined to be a natural transformation $X\to Y$. The category of presheaves $[C^o,\mathbf{Set}]$ is locally small and we shall denote it by $\mathbf{P}(C)$.

• Recall that a [simplicial set] is defined to be a presheaf on the [simplicial category] $\Delta$. We shall denote the category of simplicial sets $\mathbf{P}(\Delta^o)$ by $\mathbf{SSet}$.

• If $K\mathbf{Vect}$ is the category of $K$-vector spaces over a field $K$ and $G$ is a group, then the functor category $[\mathbf{B}G,K\mathbf{Vect}]$ is the category of a $K$-linear representations of $G$.

• Let $I$ be the category defined by the poset $[1]=\{0,1\}$, and let $i$ be the unique arrow $0\to 1$. If $\mathbf{C}$ is a category, then a functor $X:I\to \mathbf{C}$ the same thing as a morphism $x:X_0\to X_1$ in $\mathbf{C}$, where $X_0=X 0$, $X_1=X 1$ and $x=X(i)$. If $Y:I\to \mathbf{C}$ is another morphism $y:Y_0\to Y_1$, then a natural transformation $f:X\to Y$ is a commutative square

$\xymatrix{X_0\ar[d]_{x}\ar[r]^{f_0}& Y_0 \ar[d]^{y} \\ X_1 \ar[r]_{f_1} &Y_1 }$

in the category $\mathbf{C}$. The category $[I,\mathbf{C}]= \mathbf{C}^I$ is called the arrow category of $\mathbf{C}$.

Revised on March 29, 2010 at 03:26:10 by joyal